The theory is relatively well accepted since it fixes a few holes and fits with a couple other incidents, although since there's a dearth of William Henry Harrison scholars and literature its not exactly rocking to the top of citation impact rankings.

The 2014 article itself, by Dr. Philip Mackowiak who prior to his retirement was at the University of Maryland School of Medicine, was prompted by writer Jane McHugh, who had done some preliminary research into the inauguration weather (it wasn't terrible), wondered about the delay between it and the pneumonia diagnosis, and was interested in writing further. It's not firewalled and can be found here; a followup Times article by the two simplifies it for lay people and he explains the research process a bit more here. Mackowiak's an interesting guy; one of his claims to fame was doing the research that debunked the long standing canard that 98.6F was the 'normal' body temperature. He then started to get into medical history; a series of clinicopathological conferences on the medical conditions of historic figures turned into longer books that have been well received by the medical community and one is published by OUP, which meant that it's also been screened by academic medical editors. In short, he's far from a quack.

One reason the Harrison piece has not received much in the way of citation is that there are only really two modern, reasonably well researched books on Harrison. The first is on his time as a political appointee in Indiana and the Northwest Territory as well as his generalship in the War of 1812, Jefferson's Hammer by Robert Owens, and is academic; the second is popular and on the 1840 campaign, The Carnival Campaign by Ronald Schafer. The former came out in 2007 and devotes all of one sentence to Harrison's death (everything after about 1815 is covered in a single concluding chapter), and in it he unsurprisingly follows the precedent of 165 years of speculation about his inauguration causing it. This whole line of inquiry made me curious to see if he's commented on the Mackowiak piece, but a brief search didn't turn up anything.

While the second book is largely focused on the campaign - even though it's not academic, it's a useful walkthrough of the most retail marketed Presidential race until Hanna runs McKinley in 1896 - it also uses the Mackowiak piece as a reason to reexamine the weeks before Harrison's death. Harrison is in generally ok shape up until the third week of his administration. He walks back and forth to a nearby market almost every day where he gets mobbed by job seekers looking for patronage jobs; there's a story where when he's too busy to see them in the White House, they just stuff their resumes into his clothing. He also gets into a nasty spat with Henry Clay over his patronage demands:

'When Clay persisted in pushing his friends for jobs, an irritated Harrison fired back, “Mr. Clay, you forget that I am President.” Finally, Harrison wrote a note to Clay ordering him to stay away from the White House and to contact him only in writing.

Back at his room in a Washington boardinghouse, Clay angrily crumpled the note in his hand. “And it has come to this!” he said angrily. “Here is my table loaded with letters from my friends in every part of the Union, applying to me to obtain offices for them, when I have not one to give, nor influence enough to procure the appointment of a friend to the most humble position.”'

This snub would have been absolutely fascinating to see play out over the course of a Harrison administration; had Harrison lived he'd almost certainly have pushed for the rechartering of the national bank (his inaugural address is mostly a mind numbing restatement of the Whig platform), and he and Clay would have had to somehow work together on that. Even if nothing else changes - there's an argument that Harrison wouldn't have pushed for Texas annexation the way Tyler did, which might have made a significant difference - and the Civil War proceeds on course following the "irrepressible conflict" school of thought, if Clay and Harrison follow the precedent of a 20 year charter for a national bank, that meant it would have existed in 1861, and there are a number of major policy decisions in American history (like the gold standard and the concentration of banking in New York) with significant ripple effects even further down the line that might very well never have occurred.

But what Harrison being in relatively good health during this period also does is to largely debunk the pneumonia-at-inauguration myth; viral pneumonia's incubation period is far shorter, and even in the 19th century, longer incubating atypical bacterial pneumonia still probably wouldn't have killed him - especially not in the way shown by Mackowiak as he carefully walks through the etiology. Harrison doesn't present any signs of illness until 3 weeks after inauguration on March 25rd when he's caught in a rainstorm the previous day and doesn't get out of wet clothes; at that point his complaint is fatigue and gastrointestinal problems rather than pulmonary issues. When he worsens it's initially his side that's the problem, not an upper respiratory infection, which only shows up later and very much looks opportunistic rather than primary; even Harrison's physician sounds a bit unsure when discussing why his patient died.

The later illness of Polk and Taylor are fairly interesting correlations of the theory along with how the varying treatments of Harrison probably made things worse, and one thing I've run across that he doesn't mention is that the water quality provided to the White House was a subject of a future discussion between outgoing and incoming Presidents - if I remember right, I think it was Buchanan telling Lincoln which well to use. My only objection to the piece is that Mackowiak doesn't talk more about the relatively common nature of many of the toxic treatments Harrison is subjected to; laudanum was the go-to cure all for about everything for 75 years, and the 'blue mass' consisting of mercury was even more widespread as a prophylactic and treatment. Lincoln is supposed to have taken the latter routinely (although he may have stopped), and I've encountered material that suggests he was joined by at least one other President in doing so.

Anyway, there's a nice podcast by the Washington Post interviewing both Mackowiak and Library of Congress Harrison specialist Barbara Bair that might be worth a listen.

That theory comes from a 2014 article in a medical journal, Death in the White House: President William Henry Harrison's Atypical Pneumonia by Jane McHugh and Philip A. Mackowiak. The full text is available at that link.

As far as I know, the theory hasn't gotten much traction (either agreement or criticism) from academic historians: a Google Scholar search turns up only nine works that cite it, mostly other medical or biological sources, plus a compilation of Wikipedia articles about Indiana history and a paper written about historical Native American leadership which cites the McHugh and Mackowiak article for biographical details about Harrison given as context for his interactions with Tecumseh.

As the McHugh article itself states, mainstream historical scholarship has largely taken the pneumonia diagnosis by Harrison's doctor, Thomas Miller, at face value. This is consistent with what I've seen elsewhere: I recall Harrison's death being attributed to pneumonia in both Howe's "What Hath God Wrought" (part of the Oxford History of the United States series) and Freehling's "Road to Disunion Volume I: Secessionists at Bay".

McHugh and Mackowiak extensively analyse Dr. Miller's surviving case notes from a modern diagnostic perspective. They conclude that Harrison very likely did have pneumonia (they complain that Miller's notes don't detail the exam findings that lead him to diagnose pneumonia, but don't seem to dispute that part of his conclusion, which was also confirmed by two other physicians Miller brought in to consult and which could be adequately diagnosed by standard exam techniques at the time in most cases (*)). Where they disagree with Miller is that Miller appears to have considered Harrison's extensive GI symptoms and some abdominal pain Miller attributed "congestion of the liver" to be secondary to the pneumonia. McHugh and Mackowiak, analyzing Miller's notes from a modern diagnostic perspective, argue instead that Harrison's primary illness was enteric fever, causing the GI and liver symptoms, with the pneumonia likely being due to a secondary infection.

Their analysis seems well-sourced and superficially plausible to me, but as I said earlier, it doesn't seem to have gotten much in the way of analysis from a historical perspective. At a guess, this is partly due to disconnect between disciplines: I imagine very few academic historians follow medical journals as a matter of course. And perhaps partially because of a relative lack of historically significant implications of the claim if true: regardless of whether the GI symptoms or the pneumonia was the primary cause of death, Harrison still died 31 days into his term of office of a sudden natural illness, contracted right around the time of his inauguration, combining pneumonia, liver, and GI complaints. And in doing so, either way he became the first President to die in office, the succession firmly established the precedent that the VP would be a full-fledged President rather than merely an Acting President and would complete the late President's full term of office instead of being a caretaker until a special election could be arranged. Moreover, Harrison's successor Tyler repudiated most of the Whig platform on which Harrison had been elected, clashed with Whig leadership in Congress. And said Whig leadership seriously considered impeaching Tyler for abuse of power over his overtly policy-based vetoes of major Whig legislation: while vetoes based on mere policy disagreement are now routine, before Tyler it had been conventional for Presidents to only veto legislation on the basis that it was unconstitutional, corrupt, or was an appropriation of unnecessary funding to the Executive Branch. All of these things are equally true regardless of the medical details of Harrison's illness.

(*) At the time, pneumonia would have been diagnosed on the basis of the patient's symptoms (pain in the lungs, shortness of breath, fever, etc) and physical exam findings (especially crackling or bubbling sounds from the affected part of the lung on inhalation and an unusually dull sound when tapped). These signs in combination are still used to diagnose pneumonia. There are more modern diagnostic techniques used to identify pneumonia, such as chest x-rays, but these are generally used to clarify ambiguous signs from symptoms+exam or to give additional info about the nature and extent of pneumonia once it's been identified: an unambiguous finding of pneumonia based on the symptoms and exam signs a c. 1840 physician would have looked for are very unlikely to be disproven by a chest x-ray.